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The circle C1 : x2 + y2 = 3, with centre at O, intersects the parabola x2 = 2y at the point P in the first quadrant. Let the tangent to the circle C1 at P touches other two circles C2 and C3 at R2 and R3, respectively. Suppose C2 and C3 have equal radii 2√3 and centres Q2 and Q3, respectively. If Q2 and Q3 lie on the y-axis, then
- a)Q2Q3 = 12
- b)R2R3 = 4√6
- c)area of the triangle OR2R3 is 6√2
- d)area of the triangle PQ2Q3 is 4√2
Correct answer is option 'B,C'. Can you explain this answer?
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The circle C1 : x2 + y2 = 3, with centre at O, intersects the parabola...
Equation of tangent at P(√2,1) is √2 x+y-3=0
If centre of C2 at (0, α) and radius equal to 2√3
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The circle C1 : x2 + y2 = 3, with centre at O, intersects the parabola...
Let's start by finding the coordinates of point P, where the circle C1 intersects the parabola.
From the equation of the parabola x^2 = 2y, we can substitute x^2 into the equation of circle C1 to get:
(2y)^2 + y^2 = 3
4y^2 + y^2 = 3
5y^2 = 3
y^2 = 3/5
y = √(3/5)
Substituting this value of y into the equation of the parabola, we get:
x^2 = 2(√(3/5))
x^2 = 2√(3/5)
x = ±√(2√(3/5))
Since we are in the first quadrant, we take the positive value:
x = √(2√(3/5))
So the coordinates of point P are ( √(2√(3/5)), √(3/5) ).
Now let's find the equation of the tangent to circle C1 at point P.
The equation of a tangent to a circle with center (h, k) and radius r is given by:
(x - h)^2 + (y - k)^2 = r^2
Substituting the coordinates of point P into this equation, we get:
(x - √(2√(3/5)))^2 + (y - √(3/5))^2 = 3
Now let's find the coordinates of the points R2 and R3, where the tangent to circle C1 touches the other two circles C2 and C3.
Since C2 and C3 have equal radii of 2, the equation of the circles C2 and C3 can be written as:
(x - a)^2 + (y - b)^2 = 4
Substituting the equation of the tangent into the equations of C2 and C3, we get:
(x - a)^2 + (y - b)^2 = 4
(x - √(2√(3/5)))^2 + (y - √(3/5))^2 = 3
Solving these two equations simultaneously will give us the coordinates of R2 and R3.
Let's solve these equations:
(x - a)^2 + (y - b)^2 = 4
(x - √(2√(3/5)))^2 + (y - √(3/5))^2 = 3
Expanding and simplifying the first equation, we get:
x^2 - 2ax + a^2 + y^2 - 2by + b^2 = 4
Expanding and simplifying the second equation, we get:
x^2 - 2√(2√(3/5))x + 2√(3/5)x + (√(2√(3/5)))^2 + y^2 - 2√(3/5)y + (√(3/5))^2 = 3
Comparing coefficients of x and y in both equations, we get:
-2a = -2√(2√(3/5)) ⇒ a = √(2√(3
From the equation of the parabola x^2 = 2y, we can substitute x^2 into the equation of circle C1 to get:
(2y)^2 + y^2 = 3
4y^2 + y^2 = 3
5y^2 = 3
y^2 = 3/5
y = √(3/5)
Substituting this value of y into the equation of the parabola, we get:
x^2 = 2(√(3/5))
x^2 = 2√(3/5)
x = ±√(2√(3/5))
Since we are in the first quadrant, we take the positive value:
x = √(2√(3/5))
So the coordinates of point P are ( √(2√(3/5)), √(3/5) ).
Now let's find the equation of the tangent to circle C1 at point P.
The equation of a tangent to a circle with center (h, k) and radius r is given by:
(x - h)^2 + (y - k)^2 = r^2
Substituting the coordinates of point P into this equation, we get:
(x - √(2√(3/5)))^2 + (y - √(3/5))^2 = 3
Now let's find the coordinates of the points R2 and R3, where the tangent to circle C1 touches the other two circles C2 and C3.
Since C2 and C3 have equal radii of 2, the equation of the circles C2 and C3 can be written as:
(x - a)^2 + (y - b)^2 = 4
Substituting the equation of the tangent into the equations of C2 and C3, we get:
(x - a)^2 + (y - b)^2 = 4
(x - √(2√(3/5)))^2 + (y - √(3/5))^2 = 3
Solving these two equations simultaneously will give us the coordinates of R2 and R3.
Let's solve these equations:
(x - a)^2 + (y - b)^2 = 4
(x - √(2√(3/5)))^2 + (y - √(3/5))^2 = 3
Expanding and simplifying the first equation, we get:
x^2 - 2ax + a^2 + y^2 - 2by + b^2 = 4
Expanding and simplifying the second equation, we get:
x^2 - 2√(2√(3/5))x + 2√(3/5)x + (√(2√(3/5)))^2 + y^2 - 2√(3/5)y + (√(3/5))^2 = 3
Comparing coefficients of x and y in both equations, we get:
-2a = -2√(2√(3/5)) ⇒ a = √(2√(3
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The circle C1 : x2 + y2 = 3, with centre at O, intersects the parabola x2 = 2y at the point P in the firstquadrant. Let the tangent to the circle C1 at P touches other two circles C2 and C3 at R2 and R3, respectively.Suppose C2 and C3 have equal radii 2√3and centres Q2 and Q3, respectively. If Q2 and Q3 lie on the y-axis,thena)Q2Q3 = 12b)R2R3 = 4√6c)area of the triangle OR2R3 is 6√2d)area of the triangle PQ2Q3 is 4√2Correct answer is option 'B,C'. Can you explain this answer?
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The circle C1 : x2 + y2 = 3, with centre at O, intersects the parabola x2 = 2y at the point P in the firstquadrant. Let the tangent to the circle C1 at P touches other two circles C2 and C3 at R2 and R3, respectively.Suppose C2 and C3 have equal radii 2√3and centres Q2 and Q3, respectively. If Q2 and Q3 lie on the y-axis,thena)Q2Q3 = 12b)R2R3 = 4√6c)area of the triangle OR2R3 is 6√2d)area of the triangle PQ2Q3 is 4√2Correct answer is option 'B,C'. Can you explain this answer? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about The circle C1 : x2 + y2 = 3, with centre at O, intersects the parabola x2 = 2y at the point P in the firstquadrant. Let the tangent to the circle C1 at P touches other two circles C2 and C3 at R2 and R3, respectively.Suppose C2 and C3 have equal radii 2√3and centres Q2 and Q3, respectively. If Q2 and Q3 lie on the y-axis,thena)Q2Q3 = 12b)R2R3 = 4√6c)area of the triangle OR2R3 is 6√2d)area of the triangle PQ2Q3 is 4√2Correct answer is option 'B,C'. Can you explain this answer? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The circle C1 : x2 + y2 = 3, with centre at O, intersects the parabola x2 = 2y at the point P in the firstquadrant. Let the tangent to the circle C1 at P touches other two circles C2 and C3 at R2 and R3, respectively.Suppose C2 and C3 have equal radii 2√3and centres Q2 and Q3, respectively. If Q2 and Q3 lie on the y-axis,thena)Q2Q3 = 12b)R2R3 = 4√6c)area of the triangle OR2R3 is 6√2d)area of the triangle PQ2Q3 is 4√2Correct answer is option 'B,C'. Can you explain this answer?.
The circle C1 : x2 + y2 = 3, with centre at O, intersects the parabola x2 = 2y at the point P in the firstquadrant. Let the tangent to the circle C1 at P touches other two circles C2 and C3 at R2 and R3, respectively.Suppose C2 and C3 have equal radii 2√3and centres Q2 and Q3, respectively. If Q2 and Q3 lie on the y-axis,thena)Q2Q3 = 12b)R2R3 = 4√6c)area of the triangle OR2R3 is 6√2d)area of the triangle PQ2Q3 is 4√2Correct answer is option 'B,C'. Can you explain this answer? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about The circle C1 : x2 + y2 = 3, with centre at O, intersects the parabola x2 = 2y at the point P in the firstquadrant. Let the tangent to the circle C1 at P touches other two circles C2 and C3 at R2 and R3, respectively.Suppose C2 and C3 have equal radii 2√3and centres Q2 and Q3, respectively. If Q2 and Q3 lie on the y-axis,thena)Q2Q3 = 12b)R2R3 = 4√6c)area of the triangle OR2R3 is 6√2d)area of the triangle PQ2Q3 is 4√2Correct answer is option 'B,C'. Can you explain this answer? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The circle C1 : x2 + y2 = 3, with centre at O, intersects the parabola x2 = 2y at the point P in the firstquadrant. Let the tangent to the circle C1 at P touches other two circles C2 and C3 at R2 and R3, respectively.Suppose C2 and C3 have equal radii 2√3and centres Q2 and Q3, respectively. If Q2 and Q3 lie on the y-axis,thena)Q2Q3 = 12b)R2R3 = 4√6c)area of the triangle OR2R3 is 6√2d)area of the triangle PQ2Q3 is 4√2Correct answer is option 'B,C'. Can you explain this answer?.
Solutions for The circle C1 : x2 + y2 = 3, with centre at O, intersects the parabola x2 = 2y at the point P in the firstquadrant. Let the tangent to the circle C1 at P touches other two circles C2 and C3 at R2 and R3, respectively.Suppose C2 and C3 have equal radii 2√3and centres Q2 and Q3, respectively. If Q2 and Q3 lie on the y-axis,thena)Q2Q3 = 12b)R2R3 = 4√6c)area of the triangle OR2R3 is 6√2d)area of the triangle PQ2Q3 is 4√2Correct answer is option 'B,C'. Can you explain this answer? in English & in Hindi are available as part of our courses for JEE.
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Here you can find the meaning of The circle C1 : x2 + y2 = 3, with centre at O, intersects the parabola x2 = 2y at the point P in the firstquadrant. Let the tangent to the circle C1 at P touches other two circles C2 and C3 at R2 and R3, respectively.Suppose C2 and C3 have equal radii 2√3and centres Q2 and Q3, respectively. If Q2 and Q3 lie on the y-axis,thena)Q2Q3 = 12b)R2R3 = 4√6c)area of the triangle OR2R3 is 6√2d)area of the triangle PQ2Q3 is 4√2Correct answer is option 'B,C'. Can you explain this answer? defined & explained in the simplest way possible. Besides giving the explanation of
The circle C1 : x2 + y2 = 3, with centre at O, intersects the parabola x2 = 2y at the point P in the firstquadrant. Let the tangent to the circle C1 at P touches other two circles C2 and C3 at R2 and R3, respectively.Suppose C2 and C3 have equal radii 2√3and centres Q2 and Q3, respectively. If Q2 and Q3 lie on the y-axis,thena)Q2Q3 = 12b)R2R3 = 4√6c)area of the triangle OR2R3 is 6√2d)area of the triangle PQ2Q3 is 4√2Correct answer is option 'B,C'. Can you explain this answer?, a detailed solution for The circle C1 : x2 + y2 = 3, with centre at O, intersects the parabola x2 = 2y at the point P in the firstquadrant. Let the tangent to the circle C1 at P touches other two circles C2 and C3 at R2 and R3, respectively.Suppose C2 and C3 have equal radii 2√3and centres Q2 and Q3, respectively. If Q2 and Q3 lie on the y-axis,thena)Q2Q3 = 12b)R2R3 = 4√6c)area of the triangle OR2R3 is 6√2d)area of the triangle PQ2Q3 is 4√2Correct answer is option 'B,C'. Can you explain this answer? has been provided alongside types of The circle C1 : x2 + y2 = 3, with centre at O, intersects the parabola x2 = 2y at the point P in the firstquadrant. Let the tangent to the circle C1 at P touches other two circles C2 and C3 at R2 and R3, respectively.Suppose C2 and C3 have equal radii 2√3and centres Q2 and Q3, respectively. If Q2 and Q3 lie on the y-axis,thena)Q2Q3 = 12b)R2R3 = 4√6c)area of the triangle OR2R3 is 6√2d)area of the triangle PQ2Q3 is 4√2Correct answer is option 'B,C'. Can you explain this answer? theory, EduRev gives you an
ample number of questions to practice The circle C1 : x2 + y2 = 3, with centre at O, intersects the parabola x2 = 2y at the point P in the firstquadrant. Let the tangent to the circle C1 at P touches other two circles C2 and C3 at R2 and R3, respectively.Suppose C2 and C3 have equal radii 2√3and centres Q2 and Q3, respectively. If Q2 and Q3 lie on the y-axis,thena)Q2Q3 = 12b)R2R3 = 4√6c)area of the triangle OR2R3 is 6√2d)area of the triangle PQ2Q3 is 4√2Correct answer is option 'B,C'. Can you explain this answer? tests, examples and also practice JEE tests.
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